From 87fbbf33fcc2ed91cc8b8a08e1c378ef49ac723d Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sat, 24 May 2014 17:17:47 +0000
Subject: [PATCH] non-recursive alternative of llpx_sn completed !

---
 .../basic_2/substitution/cofrees.ma           |   4 -
 .../basic_2/substitution/cofrees_alt.ma       |   6 +-
 .../basic_2/substitution/lleq_alt.ma          |  33 +--
 .../basic_2/substitution/lleq_alt_rec.ma      |  54 ++++
 .../{llpx_sn_alt2.ma => llpx_sn_alt.ma}       |  39 ++-
 .../basic_2/substitution/llpx_sn_alt1.ma      | 250 ------------------
 .../basic_2/substitution/llpx_sn_alt_rec.ma   | 250 ++++++++++++++++++
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |   4 +-
 8 files changed, 349 insertions(+), 291 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt_rec.ma
 rename matita/matita/contribs/lambdadelta/basic_2/substitution/{llpx_sn_alt2.ma => llpx_sn_alt.ma} (55%)
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt1.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt_rec.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma
index 5679994e2..59b8baab8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma
@@ -30,10 +30,6 @@ interpretation
 lemma cofrees_fwd_lift: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
 /2 width=1 by/ qed-.
 
-lemma cofrees_fwd_nlift: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
-#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
-qed-.
-
 lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
                            L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
 #a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma
index ebe44cf28..7bb9c9ca7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma
@@ -19,6 +19,10 @@ include "basic_2/substitution/cofrees_lift.ma".
 
 (* Alternative definition of frees_ge ***************************************)
 
+lemma nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
+#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
+qed-.
+
 lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
                     (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
                     ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
@@ -26,7 +30,7 @@ lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ 
 #L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
 #U1 #U2 #HU12 #HU2 *
 [ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
-  * /5 width=9 by cofrees_fwd_nlift, ex5_4_intro, or_intror/
+  * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
 | * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
   * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
   lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
index a209206af..aa9bc1b27 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
@@ -12,43 +12,30 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/llpx_sn_alt1.ma".
+include "basic_2/substitution/llpx_sn_alt.ma".
 include "basic_2/substitution/lleq.ma".
 
 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
 
-(* Alternative definition ***************************************************)
+(* Alternative definition (not recursive) ***********************************)
 
 theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
-                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                           ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2
+                           I1 = I2 ∧ V1 = V2
                         ) → L1 ≡[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt1 // -HL12
+#L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
 qed.
 
-theorem lleq_ind_alt: ∀S:relation4 ynat term lenv lenv.
-                      (∀L1,L2,T,d. |L1| = |L2| → (
-                         ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                         ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                         ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2
-                      ) → S d T L1 L2) →
-                      ∀L1,L2,T,d. L1 ≡[T, d] L2 → S d T L1 L2.
-#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt1 … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
-qed-.
-
 theorem lleq_inv_alt: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
                       |L1| = |L2| ∧
-                      ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                      ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                      ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
-#L1 #L2 #T #d #H elim (llpx_sn_inv_alt1 … H) -H
+                      I1 = I2 ∧ V1 = V2.
+#L1 #L2 #T #d #H elim (llpx_sn_llpx_sn_alt … H) -H
 #HL12 #IH @conj //
 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt_rec.ma
new file mode 100644
index 000000000..258e6e9a9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt_rec.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/substitution/llpx_sn_alt_rec.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Alternative definition (recursive) ***************************************)
+
+theorem lleq_intro_alt_r: ∀L1,L2,T,d. |L1| = |L2| →
+                          (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                             ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                             ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2
+                          ) → L1 ≡[T, d] L2.
+#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt_r // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed.
+
+theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv.
+                        (∀L1,L2,T,d. |L1| = |L2| → (
+                           ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2
+                        ) → S d T L1 L2) →
+                        ∀L1,L2,T,d. L1 ≡[T, d] L2 → S d T L1 L2.
+#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
+qed-.
+
+theorem lleq_inv_alt_r: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
+                        |L1| = |L2| ∧
+                        ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                        ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
+#L1 #L2 #T #d #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt2.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
similarity index 55%
rename from matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt2.ma
rename to matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
index 8fbc6f8eb..0916edb67 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
@@ -13,23 +13,23 @@
 (**************************************************************************)
 
 include "basic_2/substitution/cofrees_alt.ma".
-include "basic_2/substitution/llpx_sn_alt1.ma".
+include "basic_2/substitution/llpx_sn_alt_rec.ma".
 
 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
 
 (* alternative definition of llpx_sn (not recursive) *)
-definition llpx_sn_alt2: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝
-                         λR,d,T,L1,L2. |L1| = |L2| ∧
-                         (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
-                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                            I1 = I2 ∧ R I1 K1 V1 V2
-                         ).
+definition llpx_sn_alt: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝
+                        λR,d,T,L1,L2. |L1| = |L2| ∧
+                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
+                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           I1 = I2 ∧ R I1 K1 V1 V2
+                        ).
 
 (* Main properties **********************************************************)
 
-theorem llpx_sn_llpx_sn_alt2: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2.
+theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
 #R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt1 … H) -H
+#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
 #HL12 #IHU @conj //
 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge … H) -H //
 [ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
@@ -39,7 +39,24 @@ theorem llpx_sn_llpx_sn_alt2: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_al
   elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
   lapply (ldrop_fwd_drop2 … HLK20) #H
   lapply (ldrop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
-  elim (IHn K10 W10 … K20 0) /3 width=6 by ldrop_fwd_rfw/ -IHn
+  elim (IHn K10 W10 … K20 0) -IHn -HL12 /3 width=6 by ldrop_fwd_rfw/
   elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
 ]
-qed.  
+qed.
+
+theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
+elim (IHU … HLK1 HLK2) /3 width=6 by nlift_frees/
+#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
+lapply (ldrop_fwd_drop2 … HLK1) #H1
+lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
+@conj [ @(ldrop_fwd_length_eq1 … H1 H2) // ] -HL12
+#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
+>(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
+#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
+lapply (ldrop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
+lapply (ldrop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
+/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt1.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt1.ma
deleted file mode 100644
index 18872b671..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt1.ma
+++ /dev/null
@@ -1,250 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* alternative definition of llpx_sn (recursive) *)
-inductive llpx_sn_alt1 (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_alt1_intro: ∀L1,L2,T,d.
-                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                         ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
-                      ) →
-                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                         ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt1 R 0 V1 K1 K2
-                      ) → |L1| = |L2| → llpx_sn_alt1 R d T L1 L2
-.
-
-(* Compact definition of llpx_sn_alt1 ****************************************)
-
-lemma llpx_sn_alt1_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
-                              (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                                 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                                 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt1 R 0 V1 K1 K2
-                              ) → llpx_sn_alt1 R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt1_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
-lemma llpx_sn_alt1_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
-                            (∀L1,L2,T,d. |L1| = |L2| → (
-                               ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                               ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                               ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt1 R 0 V1 K1 K2 & S 0 V1 K1 K2
-                            ) → S d T L1 L2) →
-                            ∀L1,L2,T,d. llpx_sn_alt1 R d T L1 L2 → S d T L1 L2.
-#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
-qed-.
-
-lemma llpx_sn_alt1_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt1 R d T L1 L2 →
-                            |L1| = |L2| ∧
-                            ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                            ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt1 R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H @(llpx_sn_alt1_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpx_sn_alt1_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt1 R d (ⓕ{I}V.T) L1 L2 →
-                            llpx_sn_alt1 R d V L1 L2 ∧ llpx_sn_alt1 R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt1_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
-/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
-qed-.
-
-lemma llpx_sn_alt1_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt1 R d (ⓑ{a,I}V.T) L1 L2 →
-                             llpx_sn_alt1 R d V L1 L2 ∧ llpx_sn_alt1 R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt1_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt1_intro_alt [1,3: normalize // ] -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
-  /3 width=9 by nlift_bind_sn, and3_intro/
-| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
-  lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
-  lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
-  elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
-  @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
-]
-qed-.
-
-(* Basic forward lemmas ******************************************************)
-
-lemma llpx_sn_alt1_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt1 R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim (llpx_sn_alt1_inv_alt … H) -H //
-qed-.
-
-lemma llpx_sn_alt1_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt1 R d (#i) L1 L2 →
-                             ∨∨ |L1| ≤ i ∧ |L2| ≤ i
-                              | yinj i < d
-                              | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
-                                                 ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
-                                                 llpx_sn_alt1 R (yinj 0) V1 K1 K2 &
-                                                 R I K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #d #i #H elim (llpx_sn_alt1_inv_alt … H) -H
-#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
-#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
-#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
-/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
-qed-.
-
-(* Basic properties **********************************************************)
-
-lemma llpx_sn_alt1_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt1 R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
-qed.
-
-lemma llpx_sn_alt1_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt1 R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
-qed.
-
-lemma llpx_sn_alt1_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt1 R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
-/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
-qed.
-
-lemma llpx_sn_alt1_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
-                         llpx_sn_alt1 R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
-lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
-/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
-qed.
-
-lemma llpx_sn_alt1_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
-                         ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
-                         llpx_sn_alt1 R 0 V1 K1 K2 → R I K1 V1 V2 →
-                        llpx_sn_alt1 R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt1_intro_alt
-[ lapply (llpx_sn_alt1_fwd_length … HK12) -HK12 #HK12
-  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
-| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
-  elim (lt_or_eq_or_gt i j) #Hij destruct
-  [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
-  | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
-    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
-  | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
-  ]
-]
-qed.
-
-lemma llpx_sn_alt1_flat: ∀R,I,L1,L2,V,T,d.
-                         llpx_sn_alt1 R d V L1 L2 → llpx_sn_alt1 R d T L1 L2 →
-                         llpx_sn_alt1 R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt1_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt1_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
-]
-qed.
-
-lemma llpx_sn_alt1_bind: ∀R,a,I,L1,L2,V,T,d.
-                         llpx_sn_alt1 R d V L1 L2 →
-                         llpx_sn_alt1 R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
-                         llpx_sn_alt1 R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt1_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt1_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem llpx_sn_lpx_sn_alt1: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt1 R d T L1 L2.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/2 width=9 by llpx_sn_alt1_sort, llpx_sn_alt1_gref, llpx_sn_alt1_skip, llpx_sn_alt1_free, llpx_sn_alt1_lref, llpx_sn_alt1_flat, llpx_sn_alt1_bind/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem llpx_sn_alt1_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt1 R d T L1 L2 → llpx_sn R d T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
-[1,3: /3 width=4 by llpx_sn_alt1_fwd_length, llpx_sn_gref, llpx_sn_sort/
-| #i #Hn #L2 #d #H lapply (llpx_sn_alt1_fwd_length … H)
-  #HL12 elim (llpx_sn_alt1_fwd_lref … H) -H
-  [ * /2 width=1 by llpx_sn_free/
-  | /2 width=1 by llpx_sn_skip/
-  | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
-  ]
-| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt1_inv_bind … H) -H
-  /3 width=1 by llpx_sn_bind/
-| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt1_inv_flat … H) -H
-  /3 width=1 by llpx_sn_flat/
-]
-qed-.
-
-(* Alternative definition of llpx_sn (recursive) ****************************)
-
-lemma llpx_sn_intro_alt1: ∀R,L1,L2,T,d. |L1| = |L2| →
-                          (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                             ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                             ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
-                          ) → llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt1_inv_lpx_sn
-@llpx_sn_alt1_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt1, and3_intro/
-qed.
-
-lemma llpx_sn_ind_alt1: ∀R. ∀S:relation4 ynat term lenv lenv.
-                        (∀L1,L2,T,d. |L1| = |L2| → (
-                           ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                           ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
-                        ) → S d T L1 L2) →
-                        ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
-#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt1 … H) -H
-#H @(llpx_sn_alt1_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt1_inv_lpx_sn, and4_intro/
-qed-.
-
-lemma llpx_sn_inv_alt1: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
-                        |L1| = |L2| ∧
-                        ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                        ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt1 … H) -H
-#H elim (llpx_sn_alt1_inv_alt … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt1_inv_lpx_sn, and3_intro/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt_rec.ma
new file mode 100644
index 000000000..2671af0eb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt_rec.ma
@@ -0,0 +1,250 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (recursive) *)
+inductive llpx_sn_alt_r (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_alt_r_intro: ∀L1,L2,T,d.
+                       (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                          ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
+                       ) →
+                       (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                          ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
+                       ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
+.
+
+(* Compact definition of llpx_sn_alt_r **************************************)
+
+lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
+                               (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                                  ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                                  ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
+                               ) → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+                             (∀L1,L2,T,d. |L1| = |L2| → (
+                                ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                                ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                                ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
+                             ) → S d T L1 L2) →
+                             ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
+#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
+                             |L1| = |L2| ∧
+                             ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                             ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                             ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
+                              llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
+                              llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+  /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+  lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+  lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+  elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+  @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
+qed-.
+
+lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
+                              ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                               | yinj i < d
+                               | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+                                                  ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+                                                  llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
+                                                  R I K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
+qed.
+
+lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
+qed.
+
+lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
+/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
+qed.
+
+lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
+                          llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
+lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
+/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
+qed.
+
+lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+                          ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+                          llpx_sn_alt_r R 0 V1 K1 K2 → R I K1 V1 V2 →
+                          llpx_sn_alt_r R d (#i) L1 L2.
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
+[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
+  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+  elim (lt_or_eq_or_gt i j) #Hij destruct
+  [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+  | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+  | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+  ]
+]
+qed.
+
+lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
+                          llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
+                          llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
+
+lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
+                          llpx_sn_alt_r R d V L1 L2 →
+                          llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+                          llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
+[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
+| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
+  #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
+  [ * /2 width=1 by llpx_sn_free/
+  | /2 width=1 by llpx_sn_skip/
+  | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+  ]
+| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
+  /3 width=1 by llpx_sn_bind/
+| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+(* Alternative definition of llpx_sn (recursive) ****************************)
+
+lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
+                           (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                              ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                              ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
+                           ) → llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
+qed.
+
+lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
+                         (∀L1,L2,T,d. |L1| = |L2| → (
+                            ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                            ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+                         ) → S d T L1 L2) →
+                         ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
+#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
+qed-.
+
+lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+                         |L1| = |L2| ∧
+                         ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                         ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                         ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index fce4f7c4b..e501a1f69 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -211,11 +211,11 @@ table {
    [ { "substitution" * } {
         [ { "lazy equivalence" * } {
              [ "fleq ( ⦃?,?,?⦄ ⋕[?] ⦃?,?,?⦄ )" "fleq_fleq" * ]
-             [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt" + "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
+             [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
           }
         ]
         [ { "lazy pointwise extension of a relation" * } {
-             [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt2" + "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_lpx_sn" * ]
+             [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt_rec" + "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_lpx_sn" * ]
           }
         ]
         [ { "pointwise union for local environments" * } {
-- 
2.39.2